Choose an integration strategy

Look for structure before integrating. A function multiplied by its inner derivative suggests substitution; a polynomial times an exponential or logarithm often suits integration by parts. Distinct linear denominator factors may suit partial fractions.

Trigonometric identities simplify squared sine or cosine. Inverse-trigonometric antiderivatives handle standard quadratic expressions. Differentiate your result to check it, and include an integration constant for an indefinite integral.

\displaystyle \int u\,dv=uv-\int v\,du,\quad \int\frac{dx}{a^2+x^2}=\frac1a\arctan\frac xa+C\ (a>0)

WORKED EXAMPLE

Evaluate $\int x e^x\,dx$ .

Choose $u=x$ , $dv=e^x dx$ ; then $du=dx$ and $v=e^x$ .
Apply parts: $xe^x-\int e^x dx$ .
The answer is $(x-1)e^x+C$ .

Assumed knowledge

Use standard antiderivatives, definite integrals and the fundamental theorem of calculus. Recall product and chain rules, exponential and logarithmic functions, and algebraic factorisation.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Choose an integration strategy

Evaluate ∫xex dx\int x e^x\,dx∫xexdx.

Put the idea to work.

Evaluate $\int x e^x\,dx$ .